Farewell (and what did we miss?)

“I believe in everything until it’s disproved. So I believe in fairies, the myths, dragons.” — John Lennon

Our tour of subgroups of non–positively curved groups has reached its end.

What did we miss? Probably lots — apologies to any aggrieved authors out there.

Here are a couple of corners of the landscape that we are conscious of having failed to explore.

• Simplicial non–positive curvature. This page by Tomasz Elsner gives an introduction and many references.
• The recent work of Daniel Wise and his collaborators on special cube complexes. This includes the papers —
  • The structure of groups with a quasi–convex hierarchy, and
  • Cubulating small-cancellation groups (with Frederic Haglund).

  (Arguably we touched upon some of the beginnings of this current hot topic.)

Most likely, there are many more.

Cannon-Thurston maps for graphs of groups

Recall from our last post that given an infinite hyperbolic subgroup of a hyperbolic group , the extension of the inclusion to a continuous is called a Cannon-Thurston map. By the continuity of , if such a map exists it’s unique.

When is quasi-convex in , clearly a Cannon-Thurston map exists: equivalent geodesics in are mapped to equivalent quasi-geodesics in . However, the general question of when a Cannon-Thurston map exists if is distorted in is still open. In this post we’ll give three partial results directed at answering this question: the original result of Cannon and Thurston, and two generalizations made by Mitra.

1. Three results

1. Cannon-Thurston:
   Given , where is a hyperbolic closed 3-manifold fibering over a circle, and , where , a closed 2-manifold, is the fiber surface, there is a Cannon-Thurston map .

2. Mitra:
   Suppose is an infinite hyperbolic normal subgroup of a hyperbolic group . Then there exists a Cannon-Thurston map .

3. Mitra:
   Let be a finite graph of groups where all vertex groups and edge groups are hyperbolic, and all injections (where is incident to ) are quasi-isometric embeddings. Suppose that is also hyperbolic. Then for any vertex , there is a Cannon-Thurston map .

2. Key lemma

Both of Mitra’s results above make use of the following key lemma.

Lemma 1. Let be a proper embedding of proper hyperbolic spaces. Suppose, given , that there is a non-negative function such that as and for all geodesic segments outside an -ball around , any geodesic in joining endpoints of is outside the -ball around (see the illustration below). Then there’s a Cannon-Thurston map .

Proof: If does not extend continuously, there exist sequences in such that and for some , but and for in .

Note that as . On the other hand, for large enough, stays uniformly close to (see the discussion of visibility in the previous post). This contradicts the existence of the function .

3. Trees of hyperbolic spaces

We’ll sketch part of Mitra’s proof of result number 3 above in the original context in which he established it: that of trees of hyperbolic spaces.

A tree of hyperbolic metric spaces is a metric space admitting a map onto a simplicial tree such that there exist constants satisfying

1. For all vertices , is path connected, rectifiable, and -hyperbolic. We denote the induced path metric by .
2. All inclusions are uniformly proper. That is, there exists a function such that for all , if , then .
3. For all edges , is path connected and rectifiable. The induced path metric is denoted .
4. For every edge there exists a function such that is an isometry onto .
5. and are -quasi-isometric embeddings into and , where connects and .

[Note: although present in Mitra’s definition, we don’t think condition 2 above is necessary. It is equivalent to

2′. There exists a function such that for any , if , then , and as .

This, however, follows from the structure of the space . Let . In order to shortcut an -geodesic from to , we can’t use a path that travels to more than other vertex spaces , since the distance between such subspaces is at least 1. The length of a path in between and is bounded (up to a constant) below by which goes to infinity like as increases.]

One example of a tree of hyperbolic spaces is the “discrete hyperbolic plane.” It’s constructed by gluing squares of side length 1 in as illustrated in the figure below.

The vertical lines correspond to vertex subspaces , and each is glued to the upper vertex subspace by an isometry and to the lower vertex subspace by a -quasi-isometry stretching the metric by a factor of 2.

Theorem 2. Let be a tree of hyperbolic metric spaces. If is hyperbolic, then there exists a Cannon-Thurston map for all .

The structure of the proof is as follows. Given a geodesic segment in , a quasiconvex set is constructed containing and satisfying the condition that if is outside an -ball around , then is outside an -ball around . As , so does . Applying the key lemma above finishes the proof.

This proof has the same structure as the proof of Mitra’s other result on normal hyperbolic subgroups. The difference lies in the construction of .

4. Constructing

Below is a morally correct, albeit simplified, description of how one constructs for a given -geodesic in stages by starting with and adding pieces of other vertex spaces one by one.

Consider the images of intersecting for every edge incident to . If the intersection is large enough (this constant depends on a number of other constants defining ), add the geodesic connecting the farthest points of the preimage of in the neighboring vertex space to . Now take the resulting geodesic segments in (some of) the spaces , where is distance 1 away from in , and apply the same procedure, without backtracking to . Propagate throughout the tree without backtracking to previously visited vertices. See the illustration below.

The bulk of Mitra’s proof consists of showing that the resulting is indeed quasi-convex in .

As an exercise, pick a in one of vertex spaces of the discrete hyperbolic plane described above, construct a , and convince yourself that it’s quasi-convex, and that it’s far from a given basepoint when itself is.

5. From trees of spaces to graphs of groups

How does Theorem 2 imply the graph-of-groups formulation? Given a graph of groups with , consider its universal cover (with the corresponding vertex and edge group assignments, so that ; see below for an example).

Taking Cayley graphs of vertex groups, and gluing them using (Cayley graphs of edge groups) and the corresponding quasi-isometric embeddings, we get a tree of hyperbolic spaces satisfying the hypotheses of Mitra’s theorem.

Boundaries of Hyperbolic Groups

In this post and the next, we will discuss Cannon-Thurston maps: (continuous) extensions of inclusions of hyperbolic groups to the boundaries . It is not evident that such maps should exist when is distorted in , but they often do nonetheless. This post will focus on defining the boundary of a hyperbolic group and establishing basic properties. We’ll take a brief look at Cannon-Thurston‘s first (distorted) example. The next post will discuss the progress of Mitra towards the following open question:

Question: If are hyperbolic groups, does there exist a Cannon-Thurston map ?

In particular, he has answered the question in the affirmative for the case of normal hyperbolic subgroups of hyperbolic groups and vertex subgroups of finite hyperbolic graphs of hyperbolic groups with quasi-isometric edge inclusion maps.

1. The boundary of a Gromov-hyperbolic space

In this section, is a geodesic space that is proper and hyperbolic. We’ll see several equivalent definitions for and the closure . References include the textbooks of Bridson-Haefliger and Ghys-de la Harpe (in French) and Kapovich-Benakli’s survey paper Boundaries of Hyperbolic Groups.

Here are two examples to keep in mind. Morally, the boundary of the hyperbolic plane should be the circle at infinity in the Poincare disk model. The boundary of the Cayley graph for should be a Cantor set, which can be viewed (under the “obvious” embedding of the Cantor set in the plane – see picture below) as the set of limit points of not in itself. Note: the Cantor set is totally disconnected, so the set of ends is the same as the boundary.

Geodesic rays will be parametrized by arclength.

1. Fix a basepoint . Then is the set of equivalence classes of geodesic rays originating at (i.e., ). Ray is equivalent to , written , if , or equivalently if the images of and are a finite Hausdorff distance from each other. [Note: for CAT(0) spaces, each equivalence class contains a single ray since the metric is convex.]
2. is defined the same way, but without requiring that rays originate at .

3. is the set of equivalence classes of quasigeodesic rays (originating at any point in the space). Here we must use only the finite Hausdorff distance as the equivalence relation. [Note: this definition has no analogue for CAT(0) spaces, where there is no quasigeodesic stability.]

These three sets are in natural bijection: As equivalence classes in (1) are contained in equivalence classes in (2) which are contained in equivalence classes in (3), we need only check that each quasigeodesic equivalence class contains a geodesic ray starting at . This follows from the Arzelà–Ascoli theorem and quasigeodesic stability. (Take a sequence of generalized geodesic rays representing the points , i.e. geodesic segments connecting the basepoint to with domain extended to by the constant map at So A subsequence converges to a geodesic ray from which is Hausdorff-close to . See the illustration below.)

To topologize these boundaries, it suffices to topologize . We do this by defining convergence: as for if they are represented by generalized rays (geodesic rays allowed) from such that every subsequence of has itself a subsequence converging to pointwise and uniformly on compact sets. It is necessary to take subsequences here because we want the sequence of blue points below with red basepoint to converge (to a single point on the boundary):

Closed sets are defined to be the sets containing all their limit points. For any fixed , a fundamental system of open sets in about is the collection of , where a generalized ray if . (The choice of does not matter since asymptotic rays from a fixed basepoint in a hyperbolic space are uniformly close: within of each other.)

4. This difficulty requiring us to take subsequences does not arise in defining the boundary of CAT(0) spaces, since the metric is convex. If is both hyperbolic and CAT(0), the various defined above are homeomorphic to the inverse limit of the closed balls as varies, induced by the projection maps to these complete convex subsets. The topology is the inverse limit topology (the coarsest topology making all the maps continuous).
5. Let denote the set of continuous functions from to with the compact-open topology. Identify with a subset of , where if is a constant map, by associating to the equivalence class of the distance to function: . The point of associated to a geodesic ray is represented by the Busemann function defined as

   

   The level sets of Busemann fuctions are called horospheres, agreeing in with the classical horospheres. A Busemann function can be viewed as a sort of “distance” function to a point of , but with close points having very small distance: near instead of near 0.

6. Points of the sequential boundary are represented by sequences in that converge to a point in . To describe this convergence, we’ll need to use the Gromov product for points :

   

   In a tree, the Gromov product represents the overlap length of the segments and , or equivalently the distance from to . In a -hyperbolic space, the Gromov product gives a good proxy for (to within some fixed multiple of , depending on the definition of -hyperbolicity used.)

   Fix a basepoint . A sequence is said to converge at infinity if as , and two such sequences converge to the same point if . (Transitivity of this relation follows from one of the many equivalent definitions of -hyperbolicity: for all .) A fundamental system of neighborhoods of a boundary point is given by bounding the Gromov overlap from below by a sequence of numbers approaching infinity.

2. Basic Properties

Most of these facts are taken from Bridson-Haefliger, chapter III.H.3.

Fact: is compact. Arzelà–Ascoli gives sequential compactness, and compactness then follows from first countability.

Fact: is visible: for every , , there is a geodesic with , . Indeed, take geodesic rays originating from a basepoint with . Since , there is a point (shown in blue) a distance more than away from . For , we can choose geodesic segments joining to , and extend them to generalized geodesic lines. By -hyperbolicity, some point of the segment is a distance less than from , which we can assume is (shown in red). Then Arzelà–Ascoli gives a subsequence of converging to a bi-infinite geodesic . By -hyperbolicity, is within of for and within of for . (See the diagram below.) So and .

Given a map between hyperbolic spaces, there is at most one way to extend it continuously to the boundary .

Fact: If is a quasi-isometric embedding of geodesic, proper, hyperbolic spaces then is a (continuous) embedding, where (using the quasigeodesic model of hyperbolic boundary.) If is a quasi-isometry, then is a homeomorphism. (These statements follow immediately from quasigeodesic stability and the description of a fundamental base of neighborhoods above.)

Example: , so is not quasi-isometric to unless .

Since the hyperbolic boundary is a quasi-isometry invariant, we can associate a boundary to a hyperbolic group .

Example: If is a closed hyperbolic surface and a closed hyperbolic 3-manifold, then and .

Example: is a Cantor set.

By the above fact, the inclusion map from an undistorted hyperbolic subgroup of a hyperbolic group extends to a continuous map (in fact, an embedding) . When is an arbitrary hyperbolic subgroup of a hyperbolic group , a continuous (not necessarily injective) extension of the inclusion to is called a Cannon-Thurston map.

3. Cannon and Thurston’s Example

Thurston proved that the mapping torus of a hyperbolic surface by a surface automorphism is a hyperbolic 3-manifold if and only if is pseudo-Anosov. In particular, there exist hyperbolic 3-manifolds where is an HNN-extension of a surface group , with

Cannon-Thurston showed the inclusion extends to a surjective map , a space-filling curve. Therefore is a distorted subgroup of . This was the first example found of a Cannon-Thurston map involving a distorted subgroup. (Mitra’s work, described in the next post, generalizes this result: he shows that there is a Cannon-Thurston map for any hyperbolic normal subgroup of a hyperbolic group.) We will sketch here why the map is surjective.

acts properly by isometries on via its inclusion in . The image of the Cannon-Thurston map is the limit set : the set of accumulation points in of any -orbit in Since is a normal subgroup, elements of act on by permuting these orbits, so they fix . Consequently, they fix the convex hull of in . Therefore contains a -orbit which is quasi-isometric to itself, so the closed set contains the whole boundary . But (The interior of a ball together with some points on its boundary is alway convex, after all.) So .

Kapovich and Short give a generalization of this argument to normal subgroups of -hyperbolic groups.

Finitely presented subgroups of hyperbolic groups of cohomological dimension 2

Suppose is a group with group ring . A projective resolution of a –module is an exact sequence

where each is a projective –module. Regard as a trivial –module. The cohomological dimension of a group is the smallest integer such that admits a projective resolution

of –modules. (If there is no such integer, the cohomological dimension is infinity.) For more details, see Cohomology of groups by Kenneth S. Brown.

In this post we shall discuss the proof of Gersten’s result that hyperbolic groups of cohomological dimension have the property that their finitely presented subgroups are also hyperbolic. This result appears in Gersten’s paper Subgroups of word hyperbolic groups in dimension 2.

Standard norms on modules

Suppose is a group and is a finitely generated –module. Let be a finitely generated free –module, with a basis , such that there is a surjective homomorphism of –modules. Then is a free –basis for . Equip with the –norm

.

We give the norm . Such a norm on is, up to linear equivalence, independent on the choices of , and — that is, if another norm is obtained from another surjective homomorphism from a different finitely generated free –module , then there is a uniform constant such that for all . So is called a standard norm.

Lemma 1. Suppose

is a short exact sequence of –modules. Assume
1) is finitely generated and equipped with a standard norm ,
2) is free and finitely generated and given an norm associated to some basis, and
3) is projective.
Then there is a retraction for (that is, a map such that is the identity on ) and a constant so that for all .

Proof. Since is projective, we know that . We choose a map such that is the identity on . Let be a finitely generated free submodule generated by a subset of the generators of which contains the image of under the map . Let . Notice that is also free, but not necessarily finitely generated. We can modify the retraction to where and . Let be a surjective homomorphism of –modules where is finitely generated, free and based.

Since is free, there is a map such that . This map can be extended to by setting the images under of the elements of be . It follows this and is supported on .

Now and are both finitely generated and free. So the map is given by a finite matrix with entries in . Therefore there exists a constant such that for all . Since the standard norm on is defined as an infimum over coset representatives, we have that for all . This completes the proof.

Theorem (Eilenberg–Ganea). Let be an arbitrary group and let . Then there exists an -dimensional -complex . If is finitely presented and of finite type, then can be taken to be finite.

If the cohomological dimension of the group is infinite, then the theorem merely asserts the existence of a -complex.

The relation module of a finitely presented group

Let be a finitely presented group. So , where is freely generated by and is the normal closure of the relations . Let the presentation –complex of this group be . The relation module for is a –module defined in the following way. The underlying abelian group is , the abelianization of R , and the –action on it is induced by the conjugation of on . Since acts trivially, the induced action by is well defined. Hence this gives us a –module structure on . This is naturally isomorphic as a module to , where is the -skeleton of the universal cover of the presentation complex of , or in other words, the Cayley graph of .

Proposition 1. If is a word hyperbolic group of cohomological dimension , then is the fundamental group of a finite aspherical –complex such that
1) the relation module is finitely generated and free as a module, and
2) the standard norm on agrees up to linear equivalence with the norm from any basis for the module .

Proof. Since has cohomological dimension , it is torsion free. Torsion free hyperbolic groups are of finite type, so is the fundamental group of a finite asherical complex. In particular, by the Eilenberg-Ganea theorem we know that is the fundamental group of a finite aspherical –complex .

We need the following lemma (page 184 of Cohomology of groups).

Lemma 2. If the cohomological dimension of a –module is , and if is an exact sequence of –modules with each projective, then is projective.

From the resolution,

and the facts that are projective and has cohomological dimension , we get from Lemma 2 that is projective.

The exact sequence

then splits. So we have an isomorphism of –modules . By definition this means that is stably free as a –module. (A –module is stably free if there exist free –modules such that as –modules.)

So we can attach a finite number of -discs trivially at the basepoint of to produce a finite -complex with fundamental group and with the relation module free as a –module. This proves the first part of the result.

The second part of the proposition follows from an application of Lemma .

We are now ready to prove our main result.

Theorem 1. Let be a finitely presented subgroup of the word hyperbolic group , where is of cohomological dimension . Then is word hyperbolic.

Proof. By Proposition , has a presentation for which the relation module is free and finitely generated.

Using this fact and Schanuel’s Lemma one can establish that for any finite presentation of , the relation module is stably free.

We can choose presentations for and such that the respective presentation complexes and satisfy . Further, by adding trivial letters and corresponding relations to the presentation for (as in a previous proof) we can assume that is free.

Let . An application of the Snake Lemma can be used to show that is projective as a –module. The short exact sequence of –modules arising from the Snake Lemma

satisfies the conditions of Proposition .

It follows that there is a retraction for , such that there is a uniform constant with the property that for all . So if , then .

Let be a circuit in and let be a van Kampen diagram for in of minimal area. The area of is , where is the element of corresponding to . The area of as a circuit in is , where now is in terms of –modules. It is clear that in as a –module is less than or equal to with as a –module. By the discussion in the preceding paragraph, we can conclude that if satisfies a linear isoperimetric inequality, then so does . Since satisfies a linear isoperimetric inequality, it must be word hyperbolic.

A finitely presented non-hyperbolic subgroup of a hyperbolic group

It is natural to inquire whether hyperbolicity is inherited by subgroups. This question is easily seen to have a negative answer; finitely generated free groups of rank at least two are hyperbolic but have subgroups, such as their commutator subgroups, which are infinitely generated and so cannot be hyperbolic since hyperbolic groups are finitely presentable. Rips constructed the first example of a finitely generated non-hyperbolic subgroup of a hyperbolic group — see this post. The next natural question, then, is whether there are finitely presentable examples.

The answer is yes. Noel Brady gave the first example in his article Branched coverings of cubical complexes and subgroups of hyperbolic groups. (He gave a further exposition of the example in his 2007 CRM notes.)

Brady’s construction, which we will explain in detail, involves cubical complexes, their branched covers, and Bestvina–Brady Morse theory. He constructs a finite branched covering of a three–dimensional cubical complex which has a hyperbolic fundamental group. Then he defines a map from the branched covering to the unit circle which lifts to a Morse function on the universal covers. Combinatorial Morse theory is used to show that the map from the branched covering to the unit circle has a kernel which is finitely presented but not of type . Hyperbolic groups are always of type , so this kernel is not hyperbolic.

Branched covers

A subcomplex of a piecewise Euclidean (PE for short) cubical complex is a branching locus if it satisfies the following two conditions:
1) For each -cell of with , we have that is a disjoint union of faces of .
2) is nonempty and connected for each cell .

A subcomplex is full if whenever a set of vertices in the subcomplex spans a simplex in the complex, the simplex is itself in the subcomplex.

Lemma 1. Let be a finite PE cubical complex and be a branching locus in . For a vertex , the link is a full subcomplex of .

A branched covering is obtained as follows:

1. Take a finite covering of .
2. Lift the piecewise Euclidean metric to this covering.
3. Take the completion of the covering with respect to this metric.

Lemma 2. If is a branched covering of a finite PE cubical complex , then is also a finite PE cubical complex and there is a natural continuous surjection . If is nonpositively curved and the branching locus is a graph, then is also nonpositively curved.

Brady’s construction

Denote the following graph by and let .

Let be the graph , which is an example of a branching locus.

Let :

The fundamental group is free of rank . As a free basis, take the six loops . (The bars indicate paths traversed in the opposite direction to their orientations.) Let be the homomorphism such that

where and are the permutations and , respectively. The essential feature of this construction is that maps all commutators of or with or to non–trivial five–cycles, since .

For , let be the projections , and . All are continuous, onto, and induce homomorphisms .

Let be a retract to . For define .

The map defines a -fold covering of . Call the completion of this covering and let be the natural map .

Why is hyperbolic

We know from Lemmas 2 that is a piecewise Euclidean nonpositively curved complex, so its universal cover will be a space. By the following two theorems, to establish that is hyperbolic, it suffices to show that the universal cover of contains no isometrically embedded flat planes.

Theorem. Suppose that a group acts properly, cocompactly, and by isometries on a –hyperbolic metric space. Then is a hyperbolic group.

Theorem. Let be a metric space which has a cocompact group of isometries, and which does not contain any isometric embeddings of the Euclidean plane. Then is a –hyperbolic metric space.

Consider the vertices in . The -skeleton of the link complex around these (missing) vertices in is a complete bipartite graph where each component has four vertices (along with some half edges i.e. edges with one end–point missing). Each loop is this graph that consists of four edges maps to the boundary of a neighborhood of a point in under exactly one of the maps or . Such a loop deformation retracts to one of the commutator loops in described above. The homotopy class of these commutator loops are mapped to nontrivial five–cycles by . Therefore in the branched cover, the link structure around these vertices is distorted, i.e. the aforementioned loops lift to –fold copies.

For any isometrically embedded flat plane in the universal cover, there is a special edge in the universal cover intersecting this plane that is not parallel to it, which maps to an edge in the branching locus. Let the intersection of the flat plane with this edge be . The link of in the universal cover can be defined in the usual way. Consider the intersection of this link with the flat plane.

The projection of to the link of one of the end points of is a well defined injective map. The projected images of the loops of are loops of more than four edges, so there is a natural “angle sum” barrier to the local embeddability of such a flat plane around this edge.

Why contains a finitely presented non–hyperbolic subgroup

The orientations of the edges in give rise to a map to the oriented unit circle mapping the edges to the cirlce respecting the orientation. This extends linearly to a map .

The composition lifts to a map from the universal cover of to . The ascending and descending links of any vertex in this Morse theory setting are –spheres, which are –connected but not –connected. This implies that the kernel of the map induced by is finitely presented but not of type , and in particular not hyperbolic. For further details of such Combinatorial Morse theory, see this earlier blog post.

We conclude that the fundamental group of is hyperbolic, but has a finitely presented subgroup which is of type but not of type , and so is not hyperbolic.

Further results

Gersten and Short proved that Brady’s subgroup satisfies a polynomial isoperimetric inequality. Its exact Dehn function remains unknown.

Brady, Clay and Dani have used a variant of the construction set out above to give a hyperbolic group with a finitely presented subgroup that contains infinitely many conjugacy classes of finite–order elements (and so is not hyperbolic). Predecessors to this example were discussed in this post.

Dehn functions of subgroups of CAT(0) groups

This is a post based on a guest lecture by Pallavi Dani.

Introduction

In this post, we will discuss Dehn functions of subgroups of non-positively curved groups. Let be a finitely presented group and let be a presentation complex for some presentation of . If is a loop in , we define the filling area of to be

Furthermore, we define the Dehn function of a finite presentation for by

In these definitions length and area are defined combinatorially, that is, by counting cells.

As we have seen in previous posts [1] and [2], the Dehn functions of hyperbolic and CAT(0) groups are quite well understood:

1. is hyperbolic if and only if , or equivalently if and only if is subquadratic. This result is due to Gromov; the absence of Dehn functions between linear and quadratic is called the “Gromov Gap”.
2. If is CAT(0) then . (On account of the Gromov gap, this means that or .)

Thus, the possibilities for Dehn functions of hyperbolic and CAT(0) groups are very limited, and it is natural to ask whether subgroups of such groups inherit this property. More specifically:

1. Which functions arise as Dehn functions of subgroups of CAT(0) groups?
2. Are there gaps in the corresponding isoperimetric spectrum?
3. If so, do they correspond to something algebraic/geometric?

Subgroups of CAT(0) groups with large Dehn functions

It is possible to construct subgroups of CAT(0) groups with “large” Dehn functions. Here, by “large” we mean strictly greater than quadratic. We will discuss a 3-step scheme for constructing such subgroups, called the Bieri doubling trick. This idea originated in the work of Bieri, and it was explored extensively by Baumslag-Bridson-Miller-Short in the context of automatic groups.

The construction involves first finding distorted subgroups of CAT(0) groups. We denote the distortion of a subgroup in by .

Step 1. Find such that is a short exact sequence, is CAT(0), and is distorted, with .

Example. Here is an example of such a pair constructed by N. Brady (for details, see section 2.3.3 of The Geometry of the Word Problem for Finitely Generated Groups). The group is isomorphic to and is defined by . The automorphism is given by and . Let be , the free group generated by and . You can see that by considering the family of words (with length on the order of in ) and noting that

The final word has ‘s in between every pair of consecutive ‘s, giving a total of about letters, and it is clearly reduced in . Hence, the distortion is at least .

To see that is CAT(0), note that is isomorphic to , where . The corresponding presentation complex built out of squares can be shown to satisfy the link condition.

Step 2. Form the double of along by amalgamating two copies of along . So .

Bridson showed that . Alternatively, see Bridson-Haefliger.

Example. In fact, for and in Brady’s example above, we can do better than Bridson’s lower bound: we have , for a proof see Theorem 6.20 in Bridson-Haefliger.

To see this, write . Now consider the following family of embedded disks in :

These disks have boundary length on the order of , and Area . Moreover, the boundary loops of these disks do not admit fillings with smaller area, since is -dimensional and aspherical. (In such spaces the embedded filling is the most efficient among the fillings of any loop.) This shows that is a lower bound on the Dehn function.

Step 3. The double embeds in . Since and are CAT(0), so is .

Example. We illustrate this for our example. Write as . Then the subgroup of is isomorphic to . (It is easy to see that the relations and , and the corresponding relations involving , are satisfied, since and commute with the other generators. Further, one can show that there are no additional relations among these generators — see Baumslag-Bridson-Miller-Short.)

Our example shows that occurs as the Dehn function of a subgroup of a CAT(0) group. By varying the pair in Step 1, one can construct other examples.

Sources for Step 1 and results.

1. Brady constructed a family of pairs , where is a CAT(0) group, and . (The example above is the case n=2.) Then the double embeds in , and . Thus, occurs as the Dehn function of a subgroup of a CAT(0) group for all integers .

2. Let be a hyperbolic surface bundle over a circle. Then one has a short exact sequence , and is a CAT(0) group. Bridson and Haefliger prove that if is a finitely generated, infinite index subgroup of a hyperbolic group then . Hence, and . In fact it is . Thus, occurs as the Dehn function of a subgroup of a CAT(0) groups.

There are other sources of examples of to use in Step 1, for example due to Barnard-Brady and Samuelson. However, these do not appear to lead to additional examples of Dehn functions. For example, it remains unknown whether for or functions growing faster than can occur.

Dehn functions of kernels

The Bestvina-Brady groups, discussed in previous posts, have been a good source of counterexamples, and so one might expect them to be candidates for subgroups of CAT(0) groups with large Dehn functions. It turns out that their Dehn functions are relatively well-behaved:

• W. Dison showed that if is a Bestvina-Brady group then .
• N. Brady constructed a Bestvina-Brady group with . The lower bound was proved in a manner similar to the example above, by constructing a sequence of embedded diagrams in the level set for , which is -dimensional and contractible. Abrams-Brady-Dani-Duchin-Young generalized this example to produce a large class of Bestvina-Brady groups whose Dehn functions attain the general upper bound of .
• For a while it was thought that Stallings group might have Dehn function of the form , where is not an integer, but Dison-Elder-Riley-Young showed that its Dehn function is in fact quadratic.

One can also consider other kernels of particular classes of homomorphisms from particular classes of CAT(0) groups. For instance, Dison studies kernels of maps from direct products of free groups to free abelian groups. He shows, for example, that the kernel of a certain surjective homomorphism has Dehn function .

Brady-Guralnik-Lee define a new class of CAT(0) groups, which they call perturbed right-angled Artin groups, and consider kernels of homomorphisms to . Using these in Step 1 of the Bieri doubling trick, they produce a class of examples of subgroups of CAT(0) groups with Dehn functions of the form , where is an integer. The kernels are of type but not , unlike the original Bieri doubles, which all have finite -dimensional ‘s. Another interesting direction is to study higher dimensional Dehn functions of CAT(0) groups and their subgroups. We won’t go into the technicalities here, but just say that the -dimensional Dehn function measures the difficulty of filling -cycles with -chains (or -spheres with -balls) in a suitable model space for .

By a result of Gromov, if is a CAT(0) group, then . (As in the case , the group attains this general upper bound.) Abrams-Brady-Dani-Duchin-Young generalized Dison’s result above and showed that for a Bestvina-Brady group , one has

Moreover, these inequalities are sharp.

Hyperbolic groups

In all the examples above, the ambient CAT(0) groups are not hyperbolic. In fact hardly anything is known about Dehn functions of subgroups of hyperbolic groups. The problem is that there is essentially only one example (due to N. Brady) of a finitely presented subgroup of a hyperbolic group which is not itself hyperbolic. (A variant of this example with infinitely many conjugacy classes of finite order elements with was discussed in this post.) The Dehn function of Brady’s example is not linear, since the group is not hyperbolic. It follows from a result of Gersten-Short that its Dehn function is bounded above by a polynomial. However, the Dehn function is not known explicitly.

Fibre products and the membership problem

Baumslag, Bridson, Miller and Short (BBMS) provide criteria for what they call the fibre products to be finitely presented. They then leverage these results to show that, amongst other things, there exists a torsion-free hyperbolic group with a fibre product subgroup , such that the membership problem for in is undecidable. To connect this to our pursuit of the distortion functions of finitely generated subgroups, we recall that a finitely generated subgroup of a finitely generated group (with solvable word problem) has a solvable membership problem if and only if the distortion function of in is bounded above by a recursive function.

Theorem 1. There exists a torsion free word hyperbolic group and a finitely presented subgroup such that there is no algorithm to decide membership of and the conjugacy problem of is unsolvable. Additionally, can be chosen to be the fundamental group of a compact negatively curved 2-complex.

The group has a solvable conjugacy problem (Gersten-Short).

1. Fibre products

The central construction of BBMS is similar to the Mihailova construction we’ve covered before. To any short exact sequence

we have the associated fibre product where

If is finitely presented and is finitely generated then is finitely generated. We can think of as the graph of the relation , and so questions about equality in are questions about membership of .

When is free and is finitely presented with undecidable word problem, has many undecidable properties (Miller), but will generally not be finitely presented in this case (see the Mihailova construction). Using a refined version of the Rips construction, encapsulated by the following theorem, allows one to get finitely presented .

1-2-3 Theorem. Suppose

is exact, and let be the associated fibre product. If is finitely generated (type ), is finitely presented ( type ), and is of type , then is finitely presented.

1.1. Type and

The major conceptual underpinning of Theorem 1 is a connection between type and . Let be an Eilenberg-Maclane space for with finite 3-skeleton. We will assume that has a single vertex. This means that its 2-skeleton, , can be identified with to a finite presentation of whose generators are given by the 1-cells and whose relations are given by the attaching maps of 2-cells in . Since is finitely generated as a -module, is finite. This can be seen by taking the attaching maps of the 3-cells as the generators of the module. This identification allows one to find a nice presentation for and thus a nice presentation for .

Obtaining this presentation is rather technical and takes a major portion of BBMS, and we omit the details. However, we will sketch their approach. Let be a presentation for a group . Let be a sequence of words of the form where and is some word over . We call such a sequence an identity sequence if is freely equal to the empty word in . An equivalence relation is given on identity sequences, and we consider the action of on these sequences given by . This action naturally induces a -action on the equivalence classes of identity sequences. We can view the identity sequences equipped with this action as a -module is isomorphic to . Thus being finitely generated as a module means that there is a finite set of identity sequences such that any identity sequence can be reduced under the equivalence relation to finitely many of these sequences in this set. This identification can then be used to determine a presentation for entirely from the presentations of and .

2. Proof of Theorem 1

Theorem 1 relies on an extension of the enhanced Rips construction.

Theorem 3 (Modified Rips Construction). There is an algorithm that associates to any finite group presentation , a compact, negatively curved, piecewise hyperbolic 2-dimensional complex and a short exact sequence

such that

1. presents the group ,
2. has a single vertex ,
3. the 2-cells of are right angled hyperbolic pentagons (each side of which crosses several 1-cells),
4. ,
5. has a finite generating set of cardinality at least 2,
6. each of the 1-cells in is the unique closed geodesic in its homotopy class,
7. the homotopy class of each is represented by one of the 1-cells of ,
8. is torsion-free,
9. and each generates its centralizer.

The first seven items follow from a construction in Bridson-Haefliger and Wise. Item (8) comes from the fact that the fundamental group of any compact non-positively curved space is torsion free. Item (9) follows from (7). is hyperbolic and torsion free, so the centralizer of every non-trivial element in is cyclic. Thus, if were a proper power, its homotopy class would not correspond to a simple closed geodesic.

Theorem follows from the next result.

Theorem 4. There exists a compact negatively curved 2-complex and a finitely presented subgroup such that

1. the membership problem for is unsolvable, and
2. has unsolvable conjugacy problem.

This theorem utilizes the existence of groups of type with unsolvable word problems. Collins and Miller constructed a group with a finite 2-dimensional and unsolvable word problem. The enhanced Rips construction is applied to to for , where is a compact negatively curved 2-complex.

Following the notation of the Rips construction, take where and the are lifts of generators of . We take for a generating set for . The 1-2-3 Theorem implies that the fibre product is finitely presented.

To see that the membership problem for is unsolvable we restrict to specific words. A word on the generators is in if and only if the same word in the is equivalent to the identity in . Thus the unsolvability of the word problem for implies the unsolvability of the membership problem for .

The unsolvability of the conjugacy problem follows from the next lemma.

Lemma 5 Let be finitely generated groups. Suppose and there exists such that . If there is no algorithm to decide membership of , then has an unsolvable conjugacy problem.

The thrust of the lemma is that, given a word in , we consider where is a generator of , and this word being conjugate to in is equivalent to being in . This lemma is applied with , as item (9) of Theorem 3 tells us that the centralizer of is , and thereby gives us unsolvability of the conjugacy problem.

3. Isomorphism problem

BBMS also addressees the isomorphism problem for subgroups of fundamental groups of non-positively curved spaces. These results take us away from subgroup distortion, but are worth noting for their own sake.

Corollary of Modified Rips construction. Let and be as above. Let . Then is the fundamental group of a compact non=-positively curved squared complex, and is thus biautomatic (Niblo-Reeves).

Proof: Re-metrize by subdividing each pentagonal face by introducing a new vertex in the middle of each face and side and using these to break the pentagons into hyperbolic quadrilaterals. These quadrilaterals are replaced by Euclidean squares without changing side length. The resulting space is still non-positively curved and the original 1-cells are still geodesics. We refine these squares until their side lengths are half that of the original 1-cells in . We attach an annulus (union of two Euclidean squares) by gluing the boundary circles to the loop representing by an isometry.

Theorem 7. There exists a non-positively curved 4-dimensional complex with biautomatic fundamental group , and a (countable) recursive class of finitely presented subgroups such that there is no algorithm to determine whether or not is isomorphic to .

The group in this theorem is a direct product where is a torsion free word hyperbolic group and is an HNN-extension of given by where generates a maximal cyclic subgroup.

Corollary 8. There exists a non-positively curved manifold of dimension 9 and a recursive class of finitely presented subgroups of such that there is no algorithm to determine homotopy equivalence between covering spaces corresponding to these subgroups.

Distortion of finitely presented subgroups of non-positively curved groups II — hydra groups

This post is written by Tim Riley. (Given that I’m writing about my own work here, it would seem awkward to blog this under the partial anonymity of the collective name “Berstein.”) The main sources for this post are my papers Hydra groups with Will Dison and Hyperbolic Hydra with Noel Brady and Will Dison. The latter is currently in preparation here. I am also using some of my talk slides on hydra groups.

Hercules versus the hydra and Ackermann’s function

For Dison and me, a hydra is a finite–length positive word on the alphabet , ,…. Hercules fights a hydra by striking off its first letter. The hydra then regenerates as follows: each remaining letter , where , becomes and the are unchanged. This process — removal of the first letter and then regeneration — repeats, with Hercules victorious when (not if!) the hydra is reduced to the empty word .

Example. Hercules defeats in five strikes:

(Each arrow represents the removal of the first letter and then regeneration.)

Proving that Hercules always wins is a fun exercise.

Perhaps surprisingly, the battles tend to be of enormous duration. Define to be the number of strikes it takes Hercules to vanquish the hydra , and for integers , , define . For example, and (another fun exercise).

What we have is a variation on Ackermann’s famous fast–growing functions , defined for integers by: and . So, in particular, , and , the –fold iterated power of .

It turns out that is essentially a version of Ackermann’s function:

Proposition. For all , .

Recursion is apparent in the battle. Suppose is a hydra ( denotes some word on ). Then in the time it takes to kill , there appear letters (and many other letters besides) after the final . Disposing of those letters then represents an additional challenge for Hercules. So the time it takes to complete that initial task of killing determines the magnitude of the remaining challenge (the hydra that has grown up beyond the in that time).

Hydra groups

The reason for our interest in this Hercules–versus–the–hydra battle is that it is suited to group theoretic manifestations, even in settings of non–positive curvature. It leads to the result that even for apparently benign and , distortion can be wild.

Define

and

Theorem (Dison and Riley). Each

• is free–by–cyclic,
• can be presented with only one defining relator,
• is ,
• and is biautomatic,

and yet is a rank– free subgroup that is distorted .

Some parts of this theorem are easy to explain. That the are free–by–cyclic is evident from their presentations. As for being one–relator groups, by re–expressing the original relations as and for (convention: ) and then eliminating and defining , one can present with one relation, a nested commutator:

That is, the relation is where is the “Engel” word defined recursively by and for .

Samuelson was the first (we think) to prove is . Our approach is to recursively define a family of words by and for . By inducting on , one can verify that after substituting for every in , the words and become freely equal for all . So the relation can be replaced by to give an alternative one–relator presentation for :

Re–expressing this presentation via for as

and then checking the link condition, one finds that the presentation 2-complex, metrized so that each 2-cell is a Euclidean square, is locally . All such groups are automatic (Gersten and Short) and, indeed, are biautomatic (Niblo and Reeves).

Open question. Is there a one–relator group with a finite–rank free subgroup (or even a finitely presented subgroup) of distortion greater than these examples?

That is free of rank is harder to prove. Our method is somewhat brutal in that we analyse combinatorial manipulations of words head-on. However, it may be of interest as a new way to recognise subgroups as being free. (I think ping–pong could not be used here as that produces non-distorted free subgroups.) We suppose we have a freely reduced word on that represents the identity. So when the ‘s are carried through the word and collected at the end (using the defining relations of to change the they pass), the result is a word on the ‘s times a power of . But as represents , this word on the ‘s must freely reduce to the empty word and the power of must be zero. We show that the free reduction that occurs amongst the ‘s, must be echoed in free–reduction amongst some of the in — this is awkward; we use a somewhat messy induction on . So, in fact, must be the empty word. It follows that there are no non-trivial relations between the and so they generate a rank– free group. The most fun part of the theorem (I think) is that the distortion of in is so large. This is where the battle between Hercules and the hydra comes in. Consider —

Question. For what is ?

Let’s look at the instance where and . One can try to convert to a word on and times a power of by introducing a to pair the first letter with , and then carry the accompanying to the end of the word by conjugating through the intervening letters; then pair off the next likewise, and repeat:

Notice that the hydra battle

is playing out within this calculation. Pairing off the first letter with a corresponds to the removal of the first letter of a hydra and conjugating a through to the right–hand end corresponds to a regeneration. There is nothing special about and here. So, as Hercules triumphs, we have:

Answer: .

This calculation, run to its conclusion, is displayed in this van Kampen diagram:

Let denote the reduced word on which represents . The diagram pictured above can be paired with its mirror image, separated by three corridors of 2–cells as shown below, to give a diagram that demonstrates the equality in of a reduced word on of length (the word along the bottom of the diagram) to a word of length . Thus the distortion is at least .

Establishing the upper bound on the distortion of in is a technical slog (it takes about half the paper!) and I won’t give any details here… anyway, the lower bound is what’s most interesting.

Incidentally, the lead to easy examples of groups with extravagantly large Dehn functions:

has Dehn function when .

Hyperbolic hydra

The more recent development is:

Theorem (N. Brady, Dison and Riley). For all , there is a hyperbolic group with a free rank– subgroup whose distortion is .

The are obtained by embellishing the . I’ll present them in two ways. The first is as a free–by–cyclic group and it’s in this guise that we will show that there is a hugely distorted subgroup. The second is via “Labelled Oriented Trees” (LOTs) and is suited for establishing hyperbolicity.

As a free–by–cyclic group, is

where acts via an automorphism

where , , and the are some words on the which we won’t specify explicitly here. So comparing with , we’ve added and the seventeen , and now acts to interchange and , except it introduces some words on the whilst doing so. The basic point of this is to avoid the that is present in whilst maintaining the pattern of interaction between the and . The action of restricted to is an automorphism.

The subgroup which we claim is free of rank– and distorted is

Our proof of its freeness is similar to that of the corresponding result for . We obtain the distortion result by mimicking the calculations for in — the words involved become interspersed with the but the do not obstruct the calculations. An alternative (and entirely explicit) presentation for is encoded in the picture below, which displays the case . [The number of is . The numbers of and do not depend on .]

The letters shown on the diagram are the generators. Two defining relations are given by 2–cells: and . All the other defining relations are encoded in the graphs: an edge labelled from a vertex labelled to a vertex labelled corresponds to a relation . I won’t explain here why these presentations both give , except to say that the result in the background is that if a 2–complex admits an –valued Morse function all of whose ascending and descending links are trees, then its fundamental group is free–by–cyclic. (See, for example, here.) You can check this holds for the link, shown below, of presentation 2–complex of the LOT presentation if we metrize the cells corresponding to commutator relations as Euclidean squares and the remaining relations as shown in the picture above. [The two grey edges in the link have length , the four green edges have length , and all other edges have length .]

The way we prove that is hyperbolic is first to check that the link is large, and so the presentation 2–complex is locally CAT(0), and then invoke the The Flat Plane Theorem to conclude that the universal cover is hyperbolic. The Flat Plane Theorem requires that there be no isometrically embedded copy of the Euclidean plane. This can be checked by first identifying which edges in the link occur in simple loops of length , and then checking that no cells have all their corners giving rise to edges in the list.

Distortion of finitely presented subgroups of non-positively curved groups I

We have seen that the Rips construction can be used to construct hyperbolic groups with finitely generated subgroups of enormous distortion — that is, not bounded above by any recursive function. However, those examples were not finitely presentable. This post and the next two will look at distortion of subgroups in non–positively curved groups when the subgroup can be finitely presented. [In fact, in almost all the examples we will examine, the heavily distorted subgroup will be a finite-rank free group.]

First a reminder —

What is distortion?

Definition. The distortion of a finitely generated subgroup in a finitely generated group , is defined by

Here and denote word metrics associated to some choices of finite generating sets. Those choices are not important: different choices lead to –equivalent distortion functions.

In this post we will give a number of examples culminating in some examples of Barnard, N. Brady, and P. Dani of CAT(-1) groups with free subgroups whose distortion outgrows any finite tower of exponentials.

Abelian subgroups of semi–hyperbolic groups

As we have seen, finitely generated abelian subgroups of semi–hyperbolic (for example, hyperbolic or CAT(0)) groups are quasi–isometrically embedded and so are undistorted — that is, their distortion functions grow like (in the sense of ). This is the Algebraic Flat Torus Theorem.

Iterated Baumslag–Solitar groups

These examples are not non–positively curved, but they pave the way for non–positively curved examples to come. In each case, the subgroup we identify as having large distortion is and this leads to the Dehn functions being at least as fast growing as the distortion functions: if a short word represents a large power in this , then represents the identity and has area at least . [It is not always the case that heavily distorted subgroups lead to large Dehn functions, but it is so for the examples that follow.] And, as we’ve previously discussed, in a group that has any right to be called non–positively curved, the Dehn function can grow no faster than quadratically.

The starting point for this family of examples is often referred to (inappropriately, as Gilbert Baumslag is known to point out!) as the Baumslag–Solitar group:

The subgroup is exponentially distorted: the lower bound is apparent from the relation

that stems from the doubling effect of when it conjugates .
The figure below, which is a Van Kampan diagram for , graphically displays this calculation.

Suppose we now distort by introducing a new letter acting by . Then the distortion of inside

grows like . [We will write for the -fold iterate of the exponential function.] This time the lower bound comes from

Iterating, we find the distortion of inside

is . [We will not go into details of the upper bounds, it suffices to say they do not harbour any great subtleties.] Here is an illustration of the calculation that leads to the 3–fold iterated exponential distortion:

This family has a limit (loosely speaking) — a one–relator group first described by Baumslag, and since analyzed by Gersten, Bernasconi (University of Utah thesis, 1994) and Platonov (Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2004, , no. 3, pp. 12–17; trans. in Moscow University Mathematics Bulletin, vol. 59 (2004), no. 3, pp. 12–17 (2005)) —

Introducing as shorthand for , we can re–express this presentation as:

and we see that conjugation by again has a doubling effect on , and that conjugates to leads to a feedback effect whereby we get huge distortion of on account of diagrams of the form shown below. To be precise, if the portion of the perimeter of this diagram that excludes the huge power of is to have length , then the tree dual to the picture must have depth and so the distortion grows like .

Hyperbolic free–by-cyclic and free–by–free groups

Perhaps, on account of the connection with 3–manifolds that are mapping tori of surface diffeomorphisms, the natural first examples of non–positively curved groups with heavily distorted subgroups are free–by–cyclic and free–by–free groups where, in each case, the automorphism giving the action on the fibre distorts it exponentially.

A CAT(0) example of such a free–by–cyclic group is relatively easy to come by:

where the generator for the factor acts by and . This is an exponentially growing automorphism — that is, has length when re–expressed as a minimal length word in and . (In fact, the length grows exactly like the Fibonacci numbers.) This group can be presented as

or equivalently as

If we metrize the presentation 2–complex so that the cell corresponding to is a Euclidean square and those corresponding to and are Euclidean equilateral triangles, all of side–length , then the link condition can be verified: one checks that the links “are large” — that is, all simple loops in the graph shown below have length at least . So the universal cover is a CAT(0)–space and the group is CAT(0).

Hyperbolic free–by–cyclic examples are a little harder to pin down. [For one thing the free group has to have rank at least three: Nielsen showed that any automorphism takes to a conjugate of itself or its inverse. It follows that has a –subgroup and so is not hyperbolic.] The existence of hyperbolic free–by–cyclic and, more generally, free–by–free examples can be derived from Bestvina–Feighn and Bestvina–Feighn–Handel. There are explicit CAT(-1) free–by–cylic examples in McCammond–Wise and CAT(-1) free–by–free examples (where the free group acting on the free fibre can have any finite rank) in N. Brady–A. Miller. All necessarily have exponentially distorted free fibres.

The Barnard–Brady–Dani / Mitra examples

We will describe groups with free subgroups that are distorted like the iterated–exponential or, faster, like . These examples combine the techniques we’ve seen above for free–by–cyclic and the iterated Baumslag–Solitar groups. The idea of uniting these ideas to give hyperbolic examples is due to Mitra. We will describe subsequent examples of Barnard, Brady and Dani that are more explicit and are also CAT(-1).

Barnard, Brady and Dani use some of the combinatorial and geometric techniques created by Wise in his work on the Rips construction. As building blocks for their groups they use

where and the are positive words on of length with the property that each two letter–word appears at most once as a subword of the . They get such explicitly by chopping up Wise’s word

which has length , and in which no appears twice as a subword. In the same manner as the free–by–cyclic example described earlier they get:

Lemma. The subgroup is free of rank– and is exponentially distorted in .

Consider metrizing the presentation –complex so that each cell is built from five right–angled hyperbolic pentagons:

One can check that the large–link condition is satisfied and so the complex is locally CAT(-1). So is CAT(-1). In fact, in the link the distance between distinct and is always at least . Accordingly, Barnard–Brady–Dani describe the free subgroup generated by the as highly convex — this will be important momentarily.

Next, pursuing an analogy with the -fold iterated Baumslag–Solitar examples, let

where is a free group of rank which is identified with the exponentially distorted subgroup in and the highly convex subgroup in . So we have exponential–upon–exponential–upon–exponential… distortion leading to being distorted in . One can build a presentation 2–complex for a presentation of by gluing together copies of the presentation –complexes we defined above for , , …, . The key point is that the ‘s subgroups being highly convex prevents the link condition from failing. In summary:

Theorem. is a CAT(-1) group and the free group generated by the ‘s in the factor is distorted in .

And, pursing an analogy with , define

where the are positive words of length , the are positive words of length , and there is no subword occurring twice among any of them. Again, find these and by carving up Wise’s word ( and will work).

Similarly to before, a locally CAT(-1) presentation 2–complex can be constructed out of 2–cells each of which is built out of five hyperbolic pentagons and the result is —

Theorem. is a CAT(-1) group and the free subgroup generated by the is distorted at least like .

Non–positively curved examples of yet more extreme (Ackermannian) distortion will be described in the next post.

The Tits alternative and non-positive curvature

The following is largely based on the paper The Tits Alternative for CAT(0) Cubical Complexes by Sageev and Wise.

1.1. The Tits alternative

A group satisfies the Tits alternative if for all subgroups , either is virtually solvable or it contains a non-abelian free group. One way to think of this is that a group satisfying the Tits alternative has either “small” subgroups (subgroups and quotients of solvable groups are solvable), or “large” subgroups (extensions by groups containing a free group contain a free group). The term originates from the work of Jacques Tits in which he showed that finitely generated linear groups have this property (later named the Tits alternative).

1.2. Known results

The following groups are known to satisfy the Tits alternative:

1. Finitely generated linear groups (Tits)
2. hyperbolic groups (Gromov, Ghys and de la Harpe)
3. (Bestvina-Feighn-Handel)
4. mapping class groups of compact surfaces (Ivanov, McCarthy)
5. some large subclasses of CAT(0) groups (Ballmann-Swiatkowski, Sageev-Wise, Xie)

Thompson’s group F is an example of a group not satisfying the Tits alternative as it contains no free subgroup of rank greater than one and isn’t virtually solvable (Bleak, for instance, constructs non-virtually solvable subgroups). Another is the Grigorchuk group. In fact any group of intermediate growth can’t satisfy the Tits alternative: it can’t have a non-abelian free subgroup (otherwise its growth would in fact be exponential), and by the Milnor–Wolf theorem, if it were virtually solvable, it would have to have a finite index nilpotent subgroup (implying polynomial growth by Gromov’s theorem) or an exponentially growing solvable subgroup (again contradicting intermediate growth).

It’s still an open question whether the Tits alternative holds for all groups, i.e. for groups which act properly and cocompactly by isometries on some space.

1.3. Hyperbolic groups

In case of hyperbolic groups, a stronger version of the Tits alternative holds:

Theorem 1. A subgroup of a hyperbolic group either

1. is finite,
2. is virtually infinite cyclic, or
3. contains a nonabelian free group.

An elementary proof of the above in the case where the subgroup is torsion free is presented on Henry Wilton’s geometric group theory blog (a precursor and inspiration for the blog you’re now reading). The general case requires additional results about hyperbolic elements and is presented (in French) in Sur les groupes hyperboliques d’après Mikhael Gromov (Theorem 37, page 157).

1.4. Groups acting on cube complexes

Theorem 2 (Sageev-Wise). Suppose acts properly on a finite dimensional cube complex and has a bound on the order of finite subgroups. Then any subgroup of either contains or is virtually abelian.

In particular, if the group acts on such an freely (and thus has no non-trivial finite subgroups), its subgroups contain or are virtually abelian.

I’ll sketch the proof of this theorem following the exposition in the original paper and add a few explanatory details.

We first prove the theorem assuming that is finitely generated.

Since is assumed to be finite dimensional, we proceed, as you may have already guessed, by induction on the dimension. If the dimension is 0, then the group must be finite (since the action is proper), so we’re done. Now we assume the theorem holds for acting on all such cube complexes of dimension less than . Assume . To apply the induction step, we will identify a proper action of a subgroup of on a lower dimensional subcomplex of . This subcomplex will be a “hyperplane” (defined next) and the subgroup acting will be its stabilizer.

A cube complex is a cubical cell complex which is simply connected and in which every link is flag. Two edges of are square equivalent if they are opposite edges of a square of . Let be an -cube (identified with the standard unit cube in ). A midcube of is an intersection of with an hyperplane parallel to a face of and containing its center. For instance, there is only one midcube in an edge, two midcubes in a square, etc.

A hyperplane in is the union of all midcubes meeting the equivalence class of an edge (under the equivalence relation generated by square equivalence). A few facts about hyperplanes:

1. A hyperplane meets each cube in at most one midcube.
2. A hyperplane separates into two components.
3. The cube structure on induces a cube structure on any hyperplane in .

1.5. Tangent: ends of groups

Let be any finitely generated group, with being the set of generators. Then the number of ends of , denoted , is the number of components of the boundary . It’s a classic fact that the number of ends can only equal 0 (when is finite), 1 (e.g. ), 2 (e.g. ), or infinity (e.g. ).

The number of ends of a finitely generated group relative to a subgroup , denoted , is the number of ends of the graph , i.e. the quotient of the Cayley graph of by the action of . Equivalently, it is the number of ends of the graph in which vertices are cosets of and vertices and are connected if and only if .

It is a famous result of Stallings that if and only if splits as an amalgamated product or an HNN-extension over a finite group. In the language of Bass-Serre theory, we have:

For relative ends, the picture is more complicated:

The one of the key theorems which embody the () implication above is the Algebraic Torus Theorem (Dunwoody-Swenson), given below. First, however, we must establish the subgroup with respect to which has multiple ends.

Theorem 3 (Sageev). Suppose , a finite dimensional cube complex, without a global fixed point. Then there is a hyperplane such that .

Theorem 4 (Algebraic Torus). Suppose is finitely generated, and , a subgroup of , is virtually polycyclic with . Then one of the following holds.

1. is virtually polycyclic.
2. There exists a short exact sequence

   

   in which is virtually polycyclic and is non-elementary fuchsian (see below).

3. splits over a virtually polycyclic group.

1.6. Tangent: fuchsian groups

A fuchsian group is a subgroup of . Thus, a fuchsian group acts discretely on the hyperbolic plane . Consider the set , for some . With respect to the hyperbolic metric on , the set has no limit points: either in , since the action is discrete, or in , since every point in is infinitely far from the boundary. However, considering the Euclidean distance inherits by being embedded as the Poincaré disk in , may have limit points in . Basic hyperbolic geometry implies that:

1. the limit points occur only in ,
2. for any other , the sets of limit points of and are identical, and
3. the cardinality of the set of limit points is 0, 1, 2 or uncountable.

A fuchsian group is non-elementary when the set of limit points is uncountable. Such a group contains .

The non-elementary fuchsian groups are really the only interesting ones: the elementary ones are either finite or virtually cyclic.

Let’s recap the proof so far. We proceed by induction, and assume the theorem true for . For of dimension , since is infinite and acts properly on , it has no global fixed points. Hence we apply Sageev’s theorem and conclude that there is a hyperplane s.t. . Let . Then acts on properly. By induction, either has an subgroup, and then so does so we’re done, or is virtually f.g. abelian. In this case, we apply the Algebraic Torus Theorem to and , and conclude that (1) is virtually polycyclic, or (2) it has a non-elementary fuchsian quotient, or (3) it splits over a virtually polycyclic group. We deal with these possibilities one by one.

If is virtually polycyclic, then we can apply the following two results of Bridson to conclude that it’s in fact virtually abelian.

Lemma 5.

1. If , a complex with finitely many shapes, then acts semi-simply with a discrete set of translation lengths.
2. If acts semi-simply and properly on a space, then any virtually solvable subgroup is virtually abelian.

(The number of shapes of a metric cell complex is the number distinct isometry classes of cells. In our case, has finitely many shapes since every cube of a given dimension is isometric and is has finite dimension. An action is semi-simple if there are points in the space realizing the translation lengths of each element. I.e., if for every there’s a point realizing ).

If has a non-elementary fuchsian quotient, which must contain an , then itself must contain an .

Finally, suppose splits over a virtually polycyclic group . Then by the first part of Bridson’s lemma above, acts semi-simply, and by the second part, must be virtually abelian. We analyze amalgamated products and HNN-extensions separately.

Suppose . If or , then contains an , by the normal form theorem for amalgamated products. If , then the Bass-Serre tree, on which acts, is homeomorphic to a line in which every edge is stabilized by a conjugate of . Thus there’s an index 2 subgroup acting by translations and . Hence , and so is virtually polycyclic and hence virtually abelian by Bridson’s lemma. This takes care of the amalgamated product case. To analyze an HNN-extension by , as well as to prove this theorem for non-finitely generated , another theorem is required.

Theorem 6 (Bridson-Haefliger). Suppose by semi-simple isometries and is . Suppose also that

1. there’s a bound on the dimension of isometrically embedded flats,
2. the set of translation lengths of is discrete at 0, and
3. there is a bound on the order of finite subgroups.

Then any ascending sequence of virtually abelian subgroups stabilizes.

Suppose , with and being the two injections of into . If and , then by the normal form theorem for HNN-extensions, has an subgroup. If , then . [Otherwise, if , consider the following subgroups of :

By the Bridson-Haefliger theorem above, this sequence must stabilize, hence .] Then , and by Bridson’s lemma, is virtually abelian.

This finishes the proof for finitely generated . If is not finitely generated, it contains an ascending, non-stabilizing sequence of finitely generated subgroups

If any contains an , we’re done. If not, then by the theorem applied to finitely generated groups, each is virtually abelian. Then conditions of the Bridson-Haefliger theorem above are satisfied, however, so a non-stabilizing ascending sequence is impossible. Hence if is not finitely generated, it contains an .